bleach, refine, expose, air
晒 centers on exposing something to the elements, especially sunlight and air, to change it: bleaching fabric, airing out, or refining by open-air treatment. The idea of exposure also extends to making something visible or public.
晒 combines 日 (sun) with 西, which likely acts as a phonetic component. The character suggests exposing something to sunlight, consistent with its meanings of bleaching and airing.
The sun 日 on the left shines on the right side 西, like spreading something out under the western sun to bleach and air it.
For サイ, imagine a sigh of relief as you air out a stuffy room: sigh -> サイ, and the sun streams in to refresh everything.
to expose (to the sun, public, danger, etc.)
bleaching; soaking
pilloried criminal exposed to public view; public scorn or humiliation; person shamed in front of others
leaving things exposed outside all night
exposure to cold
bleached cotton cloth
bleaching powder
dried bamboo; white bamboo
bugbane (Cimicifuga simplex)
bleached lining (esp. tabi); bleached interior
unbleached; undyed
shopworn goods; dead stock; shelf warmer
to fade something by overwashing it
to fix one's eyes on; to stare intently at; to study carefully (with the eyes)